Question

For three vectors \[u,v,w\] which of the following expressions is not equal to any of the remaining three
A. \[{\bf{u}} \cdot ({\bf{v}} \times {\bf{w}})\]
B. \[({\bf{v}} \times {\bf{w}}) \cdot {\bf{u}}\]
C. \[{\bf{v}} \cdot ({\bf{u}} \times {\bf{w}})\]
D. \[({\bf{u}} \times {\bf{v}}).{\bf{w}}\]

Answer 1

Hint: The formula needed to answer this question must be known before we can move on to the question itself. The guidelines for the vector dot product must be understood clearly. We must employ the idea of dot products and apply it whenever a topic involves the product of vectors. In this case, it is assumed that \[u + v + w = 0\] However, since ‘u’, ‘v’, ‘w’ are vectors, their total can only be 0 when all three vectors are linearly related or are located in the same plane.

Formula Used:The scalar triple product of three vectors a, b, and c is
\[\left( {a \times b} \right) \cdot c\]



Complete step by step solution:We have been given that there are three vectors,
\[u,v,w\]
Now, we have to determine the one of the vector that is not equal to any of the remaining three vectors.
Now, let us solve that using scalar dot product.
It is to be understood that, dot product and cross product can be switched in a scalar triple product.
\[\overrightarrow {\rm{u}} \cdot (\overrightarrow {\rm{v}} \times \overrightarrow {\rm{w}} ) = (\overrightarrow {\rm{u}} \times \overrightarrow {\rm{v}} ) \cdot \overrightarrow {\rm{w}} \]
We have been already known that the vectors can be cycled through.
\[(\vec v \times \vec w) \cdot \vec u = (\vec u \times \vec v) \cdot \vec w\]
From the above obtained result, we get
\[ \Rightarrow \vec v \cdot (\vec u \times \vec w) = - \vec u \cdot (\vec v \times \vec w)\]
Therefore, for three vectors \[u,v,w\] the expression that is not equal to any of the remaining three is
\[ - \vec u \cdot (\vec v \times \vec w)\]
Options a, b and d is \[\left[ {\begin{array}{*{20}{l}}{{\bf{u}},}&{{\bf{v}},{\bf{w}}}\end{array}} \right]\] while option \[c = - [{\bf{u}},{\bf{v}},{\bf{w}}]\]



Option ‘C’ is correct



Note: Student must be careful while doing problems involving dot product and cross product. It is important to use vectors' dot products correctly. When a modulus is specified in the question, the signs must be used with caution. On applying incorrect formulas will not yield required answer. And the concept; dot product and cross product can be switched in a scalar triple product should be kept in mind in order to get the desired answer.